From the right-hand side, Dr H. E. Schaef's chicken looked like any normal cockerel, with a bright red comb and a wattle. But from the left you would think it was a hen: its body was slighter and had plainer markings.

Even its behaviour was decidedly confused. The creature attempted to mount the other hens in the yard, yet also laid small eggs itself.

When it died, Schaef decided to prepare the bird for his table. Once the bird had been plucked, it was obvious that the right half of the skeleton was much bigger than the left. When Schaef opened the abdomen to remove the gizzards, he found both a testis and an ovary with a partially formed egg.

It was as if someone had cut a hen and a cockerel in half, and merged the two bodies seamlessly down the centre.

Keen not to waste it, Schaef proceeded to roast and eat the chicken. But once the meat had been stripped off the bones, he preserved the skeleton and passed it on to his anatomist friend Madge Thurlow Macklin. She wrote up the story in the Journal of Experimental Zoology in 1923.

Today, we call these creatures "bilateral gynandromorphs". Unlike hermaphrodites, whose blending of two sexes often begins and ends at the genitals, these animals are split across their whole bodies: male on one side, female on the other.

Nearly a century after Schaef enjoyed his strange meal, many more examples have been found. Their odd characteristics could explain some of the mysteries of sex, and how our bodies develop.

Although Schaef's account is one of the most colourful reports, sightings of male-female chimera date back hundreds of years.

Unsurprisingly, courtship for these animals sometimes presents difficulties

On 7 May 1752, a Mr M Fisher of Newgate presented the Royal Society of England with a lobster of unique appearance, with "all the parts of generation double". Since then, scientists have added crabs, silk worms, butterflies, bees, snakes and various species of bird to the list of animals that can develop into bilateral gynandromorphs.

It's impossible to say exactly how common they are. Michael Clinton at the University of Edinburgh in the UK estimates that 1 in 10,000 and 1 in 1,000,000 birds develop this way. Nobody knows what the equivalent figure would be for mammals.

Unsurprisingly, courtship for these animals sometimes presents difficulties.

In 2008, a retired high-school teacher named Robert Motz was looking out his back window in Illinois when he saw a northern cardinal whose breast was exactly half the vibrant red of a male, half the dowdy grey of a female. Eventually, his observation caught the attention of ornithologist Brian Peer at Western Illinois University in Macomb, US.

Either they are quietly shunned, or actively attacked, by their peers

"It was an incredibly fascinating and striking individual," says Peer. "If you could only see one side you would think it was male or female. It was an almost perfect split."

Together, they observed the bird on 40 separate occasions. Never once was it accompanied by a mate.

Nor did it ever make an attempt to sing a song. "Whether it was even capable of vocalising, we just don't know," says Peer.

The other birds largely seemed to ignore it. This isolation is apparently common for gynandromorphs. Either they are quietly shunned, or actively attacked, by their peers.

For a long time, many assumed that the phenomenon was down to a genetic accident after conception.

Biological sex is determined by the combination of sex chromosomes. In humans, men have an X and a Y chromosome, while women have two X chromosomes. But it works differently in other species. In chickens, for instance, the males have two Z chromosomes, while the hens have a Z and a W.

Soon, the team had found another two gynandromorphs

Crucially, a cell sometimes loses one of those chromosomes, and that has big consequences for the animal's sex.

Suppose that, while a ZW chicken embryo is developing, a single cell happens to lose the W chromosome. That cell will be lacking the genes that make it female, so it will develop masculine characteristics.

If that cell then replicates, all its descendants will also be male. Meanwhile the other cells in the embryo would still be female – potentially leading the animal to grow up as a gynandromorph.

At least, that was the theory. A few years ago, Clinton received a phone call that would cause him to reconsider this idea.

One of his colleagues had been visiting a chicken farm, and had found a gynandromorph that strongly resembled Schaef's chimera bird. "He telephoned and asked if I was interested in getting it," says Clinton. "Of course, I said yes."

The chicken was essentially formed of two, non-identical twins, fused down the centre

Soon, the team had found another two gynandromorphs, all of which showed the same, mixed characteristics.

However, when Clinton screened the chickens' genes, he found completely normal sex chromosomes across the whole chicken. Down one side they were ZW, down the other they were ZZ.

In other words, the chicken was essentially formed of two, non-identical twins, fused down the centre.

That was a pretty startling result, but at first Clinton was just disappointed at having his idea proved wrong. "Like most scientists we thought we knew the answer before the experiment," he says.

Clinton now has another idea for how gynandromorphy happens.

This apparent accident may actually be a cunning evolutionary trick gone wrong

When an egg is formed, the cell is meant to discard half its chromosomes, in a bag of DNA called the "polar body". However, in rare cases the egg may keep the polar body, as well as its own nucleus.

If both are fertilised, and the cell starts dividing, each side of the body will develop with its own genome, and its own sex.

This apparent accident may actually be a cunning evolutionary trick gone wrong.

Biologists have long known that the ratios of males and females within a population can switch depending on the environment.

During stressful times, mothers are more likely to give birth to females. They tend to be more likely to mate and pass on the mother's DNA, even when times are tough.

Some parrots can hatch 20 males or females in a row, depending on the circumstances.

Now suppose one of the mother's eggs holds onto its polar body, and therefore has two nuclei. If the mother allows each one to be fertilised, she will have a half-male-half-female embryo.

The mother could then somehow reject the unwanted sex before laying the egg, neatly controlling her offspring's sex.

However, in the rare case that the unwanted nucleus is not discarded, the result will be a gynandromorph.

At the very least, Clinton's result shows that sex develops very differently in birds and mammals.

For mammals like us, it is the sex hormones coursing through our blood that seem to be most important in determining sex.

Exploring this process may be crucial for understanding the miracle of birth and reproduction

That may explain why we don't see many gynandromorphic mammals split down the middle. No matter what the DNA of the cells says, they will all be bathed in the same hormones, and develop the same sexual characteristics.

However, the fact that both sides of a bird can develop independently, shows that it is the bird's cells themselves that control their identity and growth.

This even extends to the resulting animal's behaviour. In one study from 2003, the right (male) brain of a gynandromorph zebra finch grew a thicket of neural circuits it needed to sing courtship songs. But the left (female) side was missing these structures, despite the fact that both were exposed to the same hormones.

We still don't know whether this story applies to every creature in this strange gynandromorphic menagerie.

In a few places, humans may have accidentally made these creatures more common

Josh Jahner of the University of Nevada, Reno studies beautiful asymmetrical butterflies. He suspects double-fertilised eggs may explain them, but it's possible that other mechanisms could contribute too.

Exploring this process may be crucial for understanding the miracle of birth and reproduction.

For example, animals' bodies develop with almost perfect symmetry, but how do they manage it? Studying gynandromorphs may hold the answer.

There is one more possible explanation for gynandromorphs – or at least, for a few of them. In a few places, humans may have accidentally made these creatures more common.

In April 2015, Jahner reported a peculiar coincidence. He studies American butterflies called Lycaeides, and had never seen a single gynandromorph before the 2011 Fukushima Daiichi nuclear disaster in Japan – only to come across six in the 16 months afterwards. "And I've never found any since," he says.

Researchers found a similar abundance of gynandromorphic butterflies after the Chernobyl disaster, suggesting that a low dose of radiation may increase the chances of a gynandromorph being conceived.

"There's no way of knowing if it directly caused it or not," Jahner says, "but it's a strange coincidence."

For the moment, it is just another mystery associated with these beautiful, almost mythic-looking creatures.